Transient natural convection in a rectangular enclosure with one heated side wall

This paper describes a numerical and theoretical study of the transient natural convection heating of a two-dimensional rectangular enclosure filled with fluid. The heating is applied suddenly along one of the side walls, while the remaining three walls are maintained insulated. It is shown that the process has two distinct phases, an early period dominated by conduction and a late period dominated by convection. The scaling laws for the heat transfer rate and the effectiveness (energy storage fraction) are determined based on scale analysis. These theoretical results are confirmed by numerical experiments conducted in the domain Ra = 10³−10⁶, Pr = 7, A = 1, where Ra is the Rayleigh number based on height and initial temperature difference, Pr is the Prandtl number, and A is the height/length ratio of the enclosure. Correlations for heat transfer rate and effectiveness are constructed by comparing the theoretical scaling laws with the numerical results.     

Transient heat flux measurement of natural convection in an inclined enclosure with time-periodically-varying wall temperature

The transient natural convection in an inclined enclosure filled with water is studied experimentally for the time-periodically-varying wall temperature on one side wall and constant average temperature on the opposing side wall. This system has no temperature difference between the opposing two side walls in time-averaged sense. The temperatures of two opposing walls and the heat flux across the enclosure are measured by a heat flux meter. Based on the experimental results, the effects of time-periodically-varying wall temperature and inclined angles of the enclosure on heat transfer characteristics are studied. The experimental results show that, with the upper wall temperature oscillating, the heat flux across the enclosure is also periodically varied with time, and the net heat flux is from the lower wall to the upper wall. Numerical computations are also conducted and numerical results are qualitatively assured by the experimental measurements.     

Double diffusive natural convection in a partially heated enclosure with Soret and Dufour effects

The effect of double-diffusive natural convection of water in a partially heated enclosure with Soret and Dufour coefficients around the density maximum is studied numerically. The right vertical wall has constant temperature θ_c, while left vertical wall is partially heated θ_h, with θ_h > θ_c. The concentration in right wall is maintained higher than left wall (C_c < C_h) for case I, and concentration is lower in right wall than left wall (C_h > C_c) for case II. The remaining left vertical wall and the two horizontal walls are considered adiabatic. Water is considered as the working fluid. The governing equations are solved by control volume method using SIMPLE algorithm with QUICK scheme. The effect of the various parameters (thermal Rayleigh number, center of the heating location, density inversion parameter, Buoyancy ratio number, Schmidt number, and Soret and Dufour coefficients) on the flow pattern and heat and mass transfer has been depicted. Comprehensive Nusselt and Sherwood numbers data are presented as functions of the governing parameters mentioned above.     

Natural convection flow under a magnetic field in an inclined square enclosure differentialy heated on adjacent walls

Steady, laminar, natural-convection flow in the presence of a magnetic field in an inclined square enclosure differentially heated along the bottom and left vertical walls while the other walls are kept isothermal was considered. The governing equations were solved numerically for the stream function, vorticity and temperature ratio using the differential quadrature method for various Grashof and Hartmann numbers, inclination angle of the enclosure and direction of the magnetic field. The orientation of the enclosure changes the temperature gradient inside and has a significant effect on the flow pattern. Magnetic field suppresses the convective flow and its direction also influences the flow pattern, causing the appearance of inner loops and multiple eddies. The surface heat flux along the bottom wall is slightly increased by clockwise inclination and reduced by half by the counterclockwise inclination. The surface heat flux along the upper portion of the left side wall is reversed by the rise of warmer fluids due to the convection currents for no inclination and clockwise inclination of the enclosure.     

Laminar natural convection in a fully partitioned enclosure containing fluid with nonlinear thermophysical properties

The effects of a heat conducting partition on the laminar natural convection heat transfer and fluid flow were obtained by comparing the numerical and experimental results for a cubic enclosure without and with a partition. The two opposite vertical walls of the enclosure were isothermal at different temperatures. The working fluid was glycerol. The complete vertical partition, made of Plexiglass, was positioned in the middle of the enclosure. The visualizations of the velocity and temperature fields were obtained by using respectively, Plexiglass and liquid crystal particles as tracers. A middle plane perpendicular to the partition was numerically modeled. The steady two-dimensional model accounted for the variable thermophysical properties of the fluid. The finite volume method based on the finite difference approach was applied. The convective terms were approximated using a deferred correction central difference scheme. The velocity and temperature fields and the distribution of the local and average Nusselt numbers were found as a function of the Rayleigh (38 000 <Ra <369 000) and Prandtl (2700 < Pr < 7000) numbers.     

Double-diffusive natural convection in a porous enclosure partially heated from below and differentially salted

This paper reports numerical results of two-dimensional double-diffusive natural convection in a square porous cavity partially heated from below while its upper surface is cooled at a constant temperature. The vertical walls of the porous matrix are subjected to a horizontal concentration gradient. The parameters governing the problem are the thermal Rayleigh number (Ra=100 and 200), the Lewis number (Le=0.1, 1 and 10), the buoyancy ratio (−10N10) and the relative position of the heating element with respect to the vertical centerline of the cavity (δ=0 and 0.5). The effect of the governing parameters on fluid characteristics is analyzed. The multiplicity of solutions is explored and the existence of asymmetric bicellular flow is proved when the heated element is shifted towards a vertical boundary (δ=0.5). The solutal buoyancy forces induced by horizontal concentration gradient lead to the elimination of the multiplicity of solutions obtained in pure thermal convection when N reaches some threshold value which depends on Le and Ra.     

Numerical simulation of laminar natural convection in a laterally heated vertical cylindrical enclosure: application to crystal growth

Laminar natural convection has been studied in a laterally heated vertical cylindrical enclosure with a free insulated surface and a centrally located constant temperature wall at the top. These conditions are a simplification of the conditions existing in a Czochralski crystal pulling system. The laminar, axisymmetric flow of a Newtonian, constant physical properties fluid under Boussinesq’s approximation has been considered. Governing equations in primitive variable form are solved numerically by control volume method. SIMPLE algorithm due to Patankar has been used for the numerical simulation. The effects of the constant wall heat flux boundary condition at the side wall have been investigated whereas the bottom wall is considered to be insulated. Streamlines and isotherms are presented for various Rayleigh numbers and Prandtl numbers. Heat flux vectors through the melt are plotted for selected cases. The axial velocity and temperature variations at different horizontal sections of the crucible have been presented graphically to explain the transport processes inside the crucible. It has been observed that in case of low Pr and high Ra, flow separation occurs at the vertical wall of the crucible which leads to an oscillatory flow as Ra increases. The investigation has been extended to the oscillatory regime of flow in the zone of supercritical Rayleigh numbers and some unsteady results are also presented. Finally a heat transfer correlation has been developed for steady-state case.     

Conjugate conduction-natural convection in an enclosure with time-periodic sidewall temperature and inclination

The unsteady conjugate conduction-natural convection in enclosure is of great theoretical significance and is widely encountered in engineering applications in the areas of fluid dynamics and heat transfer. However, there are relatively few efforts to investigate the unsteady flow physics and heat transfer characteristics in the inclined enclosure of finite thickness walls. In the present work, this problem is numerically investigated by a high accuracy multidomain temporal-spatial pseudospectral method. The enclosure is filled with Boussinesq fluid and is bounded by four finite thickness and conductive walls; one of the vertical sidewall is exposed to time-periodic temperature environment while the opposite sidewall holds constant temperature; the top and bottom walls are assumed to be adiabatic. Particular efforts are focused on the effects of three types of influential factors: the wall thermophysical properties, the time-periodic temperature patterns and the inclination, and the time-periodic flow patterns and heat transfer characteristics are presented. Numerical results reveal that within the present parameter range, the heat transfer rate increases almost linearly with the thermal conductivity ratio and thermal diffusivity ratio but decreases with the inclination angle. Moreover, the heat transfer could be enhanced or weakened by selecting different temperature pulsating period in the case of finite thickness wall, while it is always enhanced if the walls are zero thickness. The back heat transfer and heat transfer resonance phenomena are observed, and their relationships with the time-periodic flow patterns and temperature distributions are analyzed. The findings are helpful to the understandings of the fluid flow and heat transfer mechanisms in the related enclosure configurations, and may be of engineering use in thermal design improvement.     

A numerical study of three-dimensional laminar natural convection in a vented enclosure

A three-dimensional numerical investigation of steady laminar natural convection in vented enclosures is carried out. A discrete flush-type heat source mounted on the substrate is used to simulate an electronic component. Four different vent locations are investigated. Combined natural convection in the air and conduction in the heat source, the substrate, and the enclosure walls are solved. Solutions are obtained for Rayleigh numbers ranging from 10⁴ to 10⁶, different substrate thermal conductivity ratios, and varied vent sizes. The calculation domain is extended beyond the cubic enclosure in x-, y-, and z-directions. Appropriate boundary conditions are prescribed on the extended computational domain. The resulting flow and temperature patterns are discussed. Also, the local and overall heat transfer from the heat source and the substrate, in terms of Nusselt numbers and the surface temperatures, are presented to illustrate the vent effects.     

MHD natural convection flow and heat transfer in a laterally heated partitioned enclosure

This study looks at MHD natural convection flow and heat transfer in a laterally heated enclosure with an off-centred partition. Governing equations in the form of vorticity–stream function formulation are solved using the polynomial differential quadrature (PDQ) method. Numerical results are obtained for various values of the partition location, Rayleigh, Prandtl and Hartmann numbers. The results indicate that magnetic field significantly suppresses flow, and thus heat transfer, especially for high Rayleigh number values. The results also show that the x-directional magnetic field is more effective in damping convection than the y-directional magnetic field, and the average heat transfer rate decreases with an increase in the distance of the partition from the hot wall. The average heat transfer rate decreases up to 80% if the partition is placed at the midpoint and an x-directional magnetic field is applied. The results also show that flow and heat transfer have little dependence on the Prandtl number.     

Free convection in tilted rectangular enclosures heated at the bottom wall and vented by different slots-venting arrangements

Free convection from a tilted rectangular enclosure heated at the bottom wall and vented by uniform slots opening at different walls of the enclosure was experimentally investigated. The experiments were carried out to study the effects of venting arrangement, opening ratio and enclosure's tilt angle on the passive cooling of the enclosure. The experiments were carried out at a constant heat flux of 250 W/m² and for enclosure tilt angles ranging from 0° to 180°. Three different venting arrangements of the air from the enclosure were studied: (1) top-venting arrangement, (2) side-venting arrangement, and (3) top and side-venting arrangement. Each venting arrangement was studied at different opening ratios of 1, 0.75, 0.5 and 0.25. The results showed that: (1) for top-venting arrangement, the Nusselt number decreases as the tilt angle of the enclosure increases, (2) for side-venting and side and top-venting arrangements, the Nusselt number increases as the tilt angle increases in the range [0°, 90°], then it decreases with the increase of the tilt angle, (3) for the three venting arrangements and at any tilt angle, the Nusselt number increases with the increase of the opening ratio of the slots, (4) for any tilt angle and at any opening ratio, the top and side-venting arrangement has the highest rate of cooling of the enclosure, and (5) for small tilt angles, the rate of cooling of the enclosure for top-venting arrangement was higher than that for side-venting arrangement, but with increasing tilt angle, the rate of cooling for side-venting arrangement becomes higher than that for top-venting arrangement. Correlations were developed for the three venting arrangements to predict the average Nusselt number of the enclosure in terms of the opening ratio and the enclosure tilt angle.     

Experiments on the cooling by natural convection of an array of vertical heated plates with constant heat flux

This paper presents experimental results that aim to document the phenomenon of cooling by natural convection of an array of vertical plates with uniform and equal heat fluxes. The working fluid is air. The effect of several factors on the plate temperature distribution was determined. These factors are the spacing of the plates, the existence of a floorlike flow obstruction near the entrance of the vertical channel, and the existence of a ceilinglike flow obstruction near the exit of the vertical channel. In several cases, these factors had a paramount effect on the plate temperature distribution. Examined also was the impact on the effectiveness of the natural convection cooling of a second row of plates positioned under the row of plates under investigation in an aligned or a staggered fashion.     

Free convection heat transfer from an isothermal wavy surface in a porous enclosure

The coupled streamfuction–temperature equations governing the Darcian flow and convection process in a fluid-saturated porous enclosure with an isothermal sinusoidal bottom sun face, has been numerically analyzed using a finite element method (FEM). No restrictions have been imposed on the geometrical non-linearity arising from the parameters like wave amplitude (a), number of waves per unit length (N), wave phase (Φ), aspect ratio (A) and also on the flow driving parameter Rayleigh number (Ra). The numerical simulations for varying values of Ra bring about interesting flow features, like the transformation of a unicellular flow to a multicellular flow. Both with increasing amplitude and increasing number of waves per unit length, owing to the shift in the separation and reattachment points, a row–column pattern of multicellular flow transforms to a simple row of multicellular flow. A cycle of n celluar and n+1 cellular flows, with the flow in adjacent cells in the opposite direction, periodically manifest with phase varying between 0 and 360°. The global heat transfer into the system has been found to decrease with increasing amplitude and increasing number of waves per unit length. Only marginal changes in the global heat flux are observed, either with increasing Ra or varying Φ. Effectively, sinusoidal bottom surface undulations of the isothermal wall of a porous enclosure reduces the heat transfer into the system. © 1998 John Wiley & Sons, Ltd.     

Lattice Boltzmann simulation of natural convection in open end cavity with inclined hot wall

Natural convection in an open end cavity with a hot inclined wall is simulated based on the lattice Boltzmann method (LBM). The physics of flow and energy transfer in open end cavities are addressed when the hot wall is inclined. The combination of the two topics (open cavity and inclined walls) is the main novelty of the present study. The effects of the angle of the hot inclined wall on the flow field and heat transfer are thoroughly investigated. The Prandtl number is fixed to 0.71 (air). The Rayleigh number and the angle of the hot inclined wall are varied in the range of 10⁴ to 10⁶ and 60? to 85?, respectively. The results are presented for two different aspect ratios, i.e., A = 1 and 2. The results obtained with the LBM are also compared with those of the finite volume method (FVM). The predicted results of the LBM conform to those of the FVM. The results show that by increasing the angle of the hot inclined wall and the aspect ratio of the cavity, the average Nusselt number decreases. The trend of the local Nusselt number on the inclined wall is also discussed.     

Conjugate natural convection of non-Newtonian hybrid nanofluid in wavy-shaped enclosure

The present study concerns the modelization and numerical simulation for the heat and flow exchange characteristics in a novel configuration saturated with a nonNewtonian Ag-MgO hybrid nanofluid. The wavy shaped enclosure is equipped with onequarter of a conducting solid cylinder. The system of equations resulting from the mathematical modeling of the physical problem in its dimensionless form is discretized via the higher-order Galerkin-based finite element method(GFEM). The dependency of vario...     

Influence of wall conduction on natural convection in an inclined square enclosure

A numerical study has been made of natural convection in an externally heated inclined enclosure with finitely conducting side walls. Results indicate that conduction along the enclosure walls has a stabilizing influence onthe convective motion in the enclosure and is therefore responsible for reduced heat transfer from the enclosure. The average Nusselt number along the hot and cold surfaces is observed to decrease with decreasing conductivity ratio and increasing wall thickness. The heat flux along the side walls is observed to reverse its direction and therefore, the Nusselt number along the side wall interfaces can be either positive or negative.

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Einfluß der Wärmeleitung in der Wand auf die freie Konvektion in einem geneigten quadratischen Raum

Zusammenfassung Es wurde eine numerische Studie der freien Konvektion in einem von außen beheizten geneigten Raum mit endlicher Wärmeleitung in den Seitenwänden durchgeführt. Die Ergebnisse zeigen, daß die Leitung längs der Raumwände einen stabilisierenden Einfluß auf die Konvektionsbewegung in dem Raum hat und deshalb für die Verringerung des Wärmeübergangs aus dem Raum verantwortlich ist. Es wurde beobachtet, daß die mittlere Nusselt-Zahl längs der warmen und kalten Oberflächen mit abnehmendem Wärmeleitverhältnis und zunehmender Wandstärke geringer wird. Es wurde auch beobachtet, daß der Wärmestrom längs der Seitenwände seine Richtung umkehrt und deshalb die Nusselt-Zahl an den Oberflächen längs der Seitenwand entweder positiv oder negativ sein kann.

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Nomenclature g gravitational acceleration - H inside height of the enclosure - k thermal conductivity of fluid - k _w thermal conductivity of enclosure walls - k _r k _w/k/Pr - L inside width of the enclosure - n coordinate direction normal to an interface - Nu Nusselt number - Nusselt number along the cold wall interface and its average value - Nusselt number along the hot wall interface and its average value - Nusselt number along the left wall interface and its average value - Nusselt number along the right wall interface and its average value - p, p*, P thermodynamic, modified and dimensionless pressures - Pr Prandtl number - Ra Rayleigh number,Ra=g (T _h – T _c ) L ³ / - T temperature - T _h ,T _c temperature of the hot and cold surfaces - T ₀ T _h + T _c )/2 - u, U dimensional and dimensionless velocity in thex-direction - U dimensionless vector velocity,i U+jV - , V dimensional and dimensionless velocity in they-direction - x,y dimensional coordinates,x along the heated and cooled walls andy along adiabatic walls - X,Y dimensionless coordinates,X=x/L, Y=y/L Greek symbols thermal diffusivity,k/( c _p ) - thermal expansion coefficient - diffusion coefficient for a general dependent variable - dimensionless fluid thermal conductivity - _s dimensionless solid thermal conductivity - kinematic viscosity - density - ₀ reference density (atT ₀) - general dependent variable - dimensionless temperature in fluid - _s dimensionless temperature in solid - angle of inclination     

Onset of unsteady axi-symmetric laminar natural convection in a vertical cylindrical enclosure heated at the wall

In the present study laminar transition to oscillatory convection of fluids having different Prandtl numbers in a laterally heated vertical cylindrical enclosure for different aspect ratios (melt height to crucible radius) of 2–4 is investigated numerically for 0.01 ≤ Pr ≤ 10. Numerical solution to two-dimensional axisymmetric transient Navier Stokes equations and energy equation were solved by finite volume method using SIMPLE algorithm. Numerical results illustrate that there exists a critical Rayleigh number for each Prandtl number beyond which sustained laminar oscillatory flow sets in. The oscillatory regime was characterised by the oscillation of the average kinetic energy and average thermal energy of the melt. For a given aspect ratio, critical Rayleigh number increases with Pr upto 1 and then flattens. It was observed that for low Prandtl number fluids, Pr < 1.0, critical Rayleigh number is found to increase with increase in aspect ratio while for high Prandtl number fluids, Pr ≥ 1.0, it is found to decrease with increase in aspect ratio. The influence of aspect ratio on the transient behaviour of the melt volume below and above the critical Rayleigh number was studied.